6th grade (Eureka Math/EngageNY)
Course: 6th grade (Eureka Math/EngageNY) > Unit 2Lesson 4: Topic D: Number theory—thinking logically about multiplicative arithmetic
- Divisibility tests for 2, 3, 4, 5, 6, 9, 10
- Recognizing divisibility
- The why of the 3 divisibility rule
- The why of the 9 divisibility rule
- Divisibility tests
- Intro to even and odd numbers
- Greatest common factor examples
- Greatest common factor explained
- Greatest common factor
- Greatest common factor review
- Least common multiple
- Least common multiple: repeating factors
- Least common multiple of three numbers
- Least common multiple
- Least common multiple review
- GCF & LCM word problems
- GCF & LCM word problems
Recognizing Divisibility. Created by Sal Khan and Monterey Institute for Technology and Education.
Want to join the conversation?
- Is zero a even number(66 votes)
- Here's a video explaining why zero is even http://www.youtube.com/watch?v=8t1TC-5OLdM(34 votes)
- Why did you skip 7 and 8?(46 votes)
- ... Why was 7 or 8 "not needed" if 3 and 6 were?
My guess is that he skipped 7 and 8 because they are quite complex. Might be easier to just try long division and see whether it works out evenly or not.
Wikipedia has more detail on how to check for 7 and 8, if you want to read: http://en.wikipedia.org/wiki/Divisibility_rule#Divisibility_rules_for_numbers_1.E2.80.9320(20 votes)
- Why does the problem skip the numbers 7 and 8? Is it like.... unsolvable?(23 votes)
- no its solvable but just more complex. seven being erase the final digit multiply it by five and add it to the rest and for eight you look at the last three digits which will become too big to recognize.(2 votes)
- So the reason we only look at the last two digits for divisibility by 4 is because they represent values that are less than 100? Because every other place value is a multiple of 100 which are divisible by 4?
Like 370 is 300 + 70. It's 3 hundreds, 7 tens and 0 ones.
300 = 100 + 100 + 100
100 / 4 = 25
So 300 / 4 = 25 * 3 = 75
But 70 is < 100
Is 70 divisible by 4?
If yes, then 370 is also divisible by 4!
Else, if no, then 370 is neither divisible by 4!
So it needs to be divisible by 2 twice?! Or 2 to the second power. But we can't use the divisibility test for 2 alone, twice. Because that would loose the purpose of the test, because we would have to divide by 2 first, if divisible by 2, and THEN test for divisibility by 2, and then divide by 2 a second time, if divisible by 2.
So to be divisible by 8 it would need to be divisible by 2 three times!? Or 2 to the third power! But again, that looses the purpose of the test. So to really test it, you need to look at the LAST THREE digits in the number?
Like 4560 is 4000 + 560.
If we test last two digits only, like before, the test will fail.
60 divisible by 8? No!
60 / 8 = 7.5
But that does NOT mean 4560 is NOT divisible by 8!
You have to test for 560! One digit or place value more than for testing with 4!
Is 560 divisible by 8?
560 / 8 = 70
So if 560 is divisible by 8, then so is 4560!
Because 4000 is a multiple of 1000.
And 1000 is a multiple of 100.
And we have already established that 100 is divisible by 4.
Then so is 1000 and 4000.
4000 = 1000 + 1000 + 1000 + 1000
But 570 is NOT a multiple of 1000!
570 < 1000
That's why we test it for divisibility by 8.
Am I getting this right?(7 votes)
- are these rules almost the same as the rules in "Divisibility Test" video?(5 votes)
- It was like the divisibility test. The question was is 730,397. was divisible by 2. No. Unless we were using decimal. Then the answer would be 365,198.50(1 vote)
- why did sal skip 7 and 8.(2 votes)
- i told you like 10 times(2 votes)
- Is there a reason 7 and 8 are not taught? I'm watching these videos to learn how to effectively teach students (currently in my last year of undergrad). Thank you!(3 votes)
- because they are more difficult than the others. As mentioned elsewhere the divisibility test for 7 is to take the last digit double it and subtract it from the remaining number. If the answer is divisible by seven the whole number is.
For eight you need to check if the last three digits are divisible by eight.
These tests are more difficult to use and aren't as applicable as the previous tests as larger numbers are required.(3 votes)
- Divisibility?What's Divisibility?(2 votes)
- Basically when we test divisibility we want to know if a number if divisible by another number without leaving any remainder.
So for example 6 = 3 x 2 so we can say 6 is divisible by 2 and 6 is also divisible by 3.
This means when we divide 6 by 2 there is no remainder left over. And the same is true if we divide 6 by 3 leaving no remainder.
There are a whole bunch of rules to test divisibility. The easiest rule is that all even numbers are divisible by 2.
Once you are comfortable with that then its worth looking into other divisibility tests for divisibility by 3, 5 and 10. Then we can look at 4 and 6 and 8.(4 votes)
- Wait so is 8 and 7 divisible by 380?(3 votes)
- no its not check it(1 vote)
- Isn't it also true that any number divisible by 2 would also be divisible by 4? If this is true, why is it necessary to have a rule specific to the number 4? Also, is it true that an odd number will never be divisible by an even number?(1 vote)
- "Isn't it also true that any number divisible by 2 would also be divisible by 4?"
No, that is false. If we used 6 as an example, it is divisible by 2 but not 4.
"Also, is it true that an odd number will never be divisible by an even number?"
Yes, that is true.(3 votes)
Determine whether 380 is divisible by 2, 3, 4, 5, 6, 9 or 10. They skipped 7 and 8 so we don't have to worry about those. So let's think about 2. So are we divisible by 2? Let me write the 2 here. Well, in order for something to be divisible by 2, it has to be an even number, and to be an even number, your ones digit-- so let me rewrite 380. To be even, your ones digit has to be even, so this has to be even. And for this to be even, it has to be 0, 2, 4, 6 or 8, and this is 0, so 380 is even, which means it is divisible by 2, so it works with 2. So 2 works out. Let's think about the situation for 3. Now, a quick way to think about 3-- so let me write just 3 question mark-- is to add the digits of your number. And if the sum that you get is divisible by 3, then you are divisible by 3. So let's try to do that. So 380, let's add the digits. 3 plus 8 plus 0 is equal to-- 3 plus 8 is 11 plus 0, so it's just 11. And if you have trouble figuring out whether this is divisible by 3, you could then just add these two numbers again, so you can actually add the 1 plus 1 again, and you would get a 2. Regardless of whether you look at the 1 or the 2, neither of these are divisible by 3. So not divisible by 3, and maybe in a future video, I'll explain why this works, and maybe you want to think about why this works. So these aren't divisible by 3, so 380 is not divisible. 380, not divisible by 3, so 3 does not work. We are not divisible by 3. Now, I'll think about the situation for 4, so we're thinking about 4 divisibility. So let me write it in orange. So we are wondering about 4. Now, something you may or may not already realize is that 100 is divisible by 4. It goes evenly. So this is 380. So the 300 is divisible by 4, so we just have to figure out whether the leftover, whether the 80, is divisible by 4. Another way to think about it is are the last two digits divisible by 4? And this comes from the fact that 100 is divisible by 4, so everything, the hundreds place or above, it's going to be divisible by 4. You just have to worry about the last part. So in this situation, is 80 divisible by 4? Now, you could eyeball that. You could say, well, 8 is definitely divisible by 4. 8 divided by 4 is 2. 80 divided by 4 is 20, so this works. Yes! Yes! So since 80 is divisible by 4, 380 is also divisible by 4, so 4 works. So let's do 5. I'll actually scroll down a little bit. Let's try 5. So what's the pattern when something is divisible by 5? Let's do the multiple of 5? 5, 10, 15, 20, 25. So if something's divisible by 5-- I could keep going-- that means it ends with either a 5 or a 0, right? Every multiple of 5 either has a 5 or a 0 in the ones place. Now 380 has a 0 in the ones place, so it is divisible by 5. Now, let's think about the situation for 6. Let's think about what happens with 6. So we want to know are we divisible by 6? So to be divisible by 6, you have to be divisible by the things that make up 6. Remember, 6 is equal to 2 times 3. So if you're divisible by 6, that means you are divisible by 2 and you are divisible by 3. If you're divisible by both 2 and 3, you'll be divisible by 6. Now, 380 is divisible by 2, but we've already established that it is not divisible by 3. If it's not divisible by 3, it cannot be divisible by 6, so this gets knocked out. We are not divisible by 6. Now, let's go to 9. So divisibility by 9. So you can make a similar argument here that if something is not divisible by 3, there's no way it's going to be divisible by 9 because 9 is equal to 3 times 3. So to be divisible by 9, you have to be divisible by 3 at least twice. At least two 3's have to go into your number, and this isn't the case, so you could already knock 9 out. But if we didn't already know that we're not divisible by 3, the other way to do it is a very similar way to figure out divisibility by 3. We can add the digits. So you add 3 plus 8 plus 0, and you get 11. And you say is this divisible by 9? And you say this is not divisible by 9, so 380 must not be divisible by 9. And for 3, you do the same thing, but you test whether the sum is divisible by 3. For 9, you test whether it's divisible by 9. So lastly, we have the number 10. We have the number 10, and this is on some level the easiest one. What do all the multiples of 10 look like? 10, 20, 30, 40, we could just keep going on and on. They all end with zero. Or if something ends with zero, it is divisible by 10. 380 does end with zero, or its ones place does have a zero on it, so it is divisible by 10. So we're divisible by all of these numbers except for 3, 6 and 9.